Termination w.r.t. Q proof of /tmp/tmpbBdP9I/OvConsOS_nokinds

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

Q restricted rewrite system:
The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(X) → X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(X) → X

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

U21¹(tt) → NIL
ISNATILIST(V) → ACTIVATE(V)
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U31¹(tt, IL, M, N) → ACTIVATE(IL)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
ACTIVATE(n__0) → 0¹
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U31¹(tt, IL, M, N) → ACTIVATE(M)
LENGTH(nil) → 0¹
ACTIVATE(n__zeros) → ZEROS
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__nil) → NIL
TAKE(0, IL) → ISNATILIST(IL)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N)))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
U31¹(tt, IL, M, N) → CONS(activate(N), n__take(activate(M), activate(IL)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
TAKE(s(M), cons(N, IL)) → U31¹(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
U11¹(tt, L) → ACTIVATE(L)
TAKE(0, IL) → U21¹(isNatIList(IL))
LENGTH(cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__take(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
U11¹(tt, L) → LENGTH(activate(L))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ACTIVATE(n__isNat(X)) → ISNAT(X)
U11¹(tt, L) → S(length(activate(L)))
ZEROS → CONS(0, n__zeros)
U31¹(tt, IL, M, N) → ACTIVATE(N)
ZEROS → 0¹
ISNAT(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

U21¹(tt) → NIL
ISNATILIST(V) → ACTIVATE(V)
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U31¹(tt, IL, M, N) → ACTIVATE(IL)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
ACTIVATE(n__0) → 0¹
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U31¹(tt, IL, M, N) → ACTIVATE(M)
LENGTH(nil) → 0¹
ACTIVATE(n__zeros) → ZEROS
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__nil) → NIL
TAKE(0, IL) → ISNATILIST(IL)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N)))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
U31¹(tt, IL, M, N) → CONS(activate(N), n__take(activate(M), activate(IL)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
TAKE(s(M), cons(N, IL)) → U31¹(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
U11¹(tt, L) → ACTIVATE(L)
TAKE(0, IL) → U21¹(isNatIList(IL))
LENGTH(cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__take(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
U11¹(tt, L) → LENGTH(activate(L))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ACTIVATE(n__isNat(X)) → ISNAT(X)
U11¹(tt, L) → S(length(activate(L)))
ZEROS → CONS(0, n__zeros)
U31¹(tt, IL, M, N) → ACTIVATE(N)
ZEROS → 0¹
ISNAT(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 12 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U31¹(tt, IL, M, N) → ACTIVATE(IL)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
TAKE(s(M), cons(N, IL)) → U31¹(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
U31¹(tt, IL, M, N) → ACTIVATE(M)
U11¹(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ACTIVATE(L)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ISNATLIST(n__take(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
U11¹(tt, L) → LENGTH(activate(L))
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U31¹(tt, IL, M, N) → ACTIVATE(N)
ISNAT(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
U11¹(tt, L) → ACTIVATE(L)
LENGTH(cons(N, L)) → ACTIVATE(L)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))

The remaining pairs can at least be oriented weakly.

ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U31¹(tt, IL, M, N) → ACTIVATE(IL)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
TAKE(s(M), cons(N, IL)) → U31¹(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
U31¹(tt, IL, M, N) → ACTIVATE(M)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ISNATLIST(n__take(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
U11¹(tt, L) → LENGTH(activate(L))
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U31¹(tt, IL, M, N) → ACTIVATE(N)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNAT(x₁)) = x₁
POL(ISNATILIST(x₁)) = x₁
POL(ISNATLIST(x₁)) = x₁
POL(LENGTH(x₁)) = 1 + x₁
POL(TAKE(x₁, x₂)) = x₁ + x₂
POL(U11(x₁, x₂)) = 1 + x₂
POL(U11¹(x₁, x₂)) = 1 + x₂
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄
POL(U31¹(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + x₁
POL(n__0) = 0
POL(n__and(x₁, x₂)) = x₁ + x₂
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__isNat(x₁)) = x₁
POL(n__isNatIList(x₁)) = x₁
POL(n__isNatList(x₁)) = x₁
POL(n__length(x₁)) = 1 + x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__take(x₁, x₂)) = x₁ + x₂
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = x₁ + x₂
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

isNatIList(X) → n__isNatIList(X)
cons(X1, X2) → n__cons(X1, X2)
isNatList(X) → n__isNatList(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
take(0, IL) → U21(isNatIList(IL))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
0 → n__0
take(X1, X2) → n__take(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
activate(X) → X
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatIList(V) → isNatList(activate(V))
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
isNat(n__s(V1)) → isNat(activate(V1))
and(tt, X) → activate(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
activate(n__isNatIList(X)) → isNatIList(X)
isNat(n__length(V1)) → isNatList(activate(V1))
activate(n__isNatList(X)) → isNatList(X)
activate(n__nil) → nil
isNat(n__0) → tt
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt) → nil
U11(tt, L) → s(length(activate(L)))
zeros → cons(0, n__zeros)
length(nil) → 0
isNatList(n__nil) → tt
isNatIList(n__zeros) → tt

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U31¹(tt, IL, M, N) → ACTIVATE(IL)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
TAKE(s(M), cons(N, IL)) → U31¹(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
U31¹(tt, IL, M, N) → ACTIVATE(M)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ISNATLIST(n__take(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
U11¹(tt, L) → LENGTH(activate(L))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N)))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U31¹(tt, IL, M, N) → ACTIVATE(N)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ Narrowing

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U11¹(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U11¹(tt, L) → LENGTH(activate(L)) at position [0] we obtained the following new rules:

U11¹(tt, n__zeros) → LENGTH(zeros)
U11¹(tt, n__length(x0)) → LENGTH(length(activate(x0)))
U11¹(tt, n__nil) → LENGTH(nil)
U11¹(tt, x0) → LENGTH(x0)
U11¹(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U11¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U11¹(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U11¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))
U11¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U11¹(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U11¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U11¹(tt, n__0) → LENGTH(0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U11¹(tt, n__nil) → LENGTH(nil)
U11¹(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U11¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U11¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U11¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U11¹(tt, n__zeros) → LENGTH(zeros)
U11¹(tt, n__length(x0)) → LENGTH(length(activate(x0)))
U11¹(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11¹(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U11¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))
U11¹(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U11¹(tt, n__0) → LENGTH(0)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U11¹(tt, n__zeros) → LENGTH(zeros) at position [0] we obtained the following new rules:

U11¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U11¹(tt, n__zeros) → LENGTH(n__zeros)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U11¹(tt, n__nil) → LENGTH(nil)
U11¹(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U11¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U11¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U11¹(tt, n__zeros) → LENGTH(n__zeros)
U11¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U11¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U11¹(tt, n__length(x0)) → LENGTH(length(activate(x0)))
U11¹(tt, x0) → LENGTH(x0)
U11¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))
U11¹(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11¹(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U11¹(tt, n__0) → LENGTH(0)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U11¹(tt, n__nil) → LENGTH(nil)
U11¹(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U11¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U11¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U11¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U11¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U11¹(tt, n__length(x0)) → LENGTH(length(activate(x0)))
U11¹(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11¹(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U11¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))
U11¹(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U11¹(tt, n__0) → LENGTH(0)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U11¹(tt, n__nil) → LENGTH(nil) at position [0] we obtained the following new rules:

U11¹(tt, n__nil) → LENGTH(n__nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U11¹(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U11¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U11¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U11¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U11¹(tt, n__nil) → LENGTH(n__nil)
U11¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U11¹(tt, n__length(x0)) → LENGTH(length(activate(x0)))
U11¹(tt, x0) → LENGTH(x0)
U11¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))
U11¹(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11¹(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U11¹(tt, n__0) → LENGTH(0)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U11¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U11¹(tt, n__length(x0)) → LENGTH(length(activate(x0)))
U11¹(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U11¹(tt, x0) → LENGTH(x0)
U11¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))
U11¹(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U11¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11¹(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U11¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U11¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U11¹(tt, n__0) → LENGTH(0)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U11¹(tt, n__0) → LENGTH(0) at position [0] we obtained the following new rules:

U11¹(tt, n__0) → LENGTH(n__0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U11¹(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U11¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U11¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U11¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U11¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U11¹(tt, n__length(x0)) → LENGTH(length(activate(x0)))
U11¹(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11¹(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U11¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))
U11¹(tt, n__0) → LENGTH(n__0)
U11¹(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U11¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U11¹(tt, n__length(x0)) → LENGTH(length(activate(x0)))
U11¹(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U11¹(tt, x0) → LENGTH(x0)
U11¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))
U11¹(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U11¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11¹(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U11¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U11¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

U11¹(tt, n__length(x0)) → LENGTH(length(activate(x0)))

The remaining pairs can at least be oriented weakly.

U11¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U11¹(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U11¹(tt, x0) → LENGTH(x0)
U11¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))
U11¹(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U11¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11¹(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U11¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U11¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( U31(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	1	1	\
\	0	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( n__s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	1	1	/

x₁

M( n__nil ) =

/	0	\
\	0	/

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	1	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( nil ) =

/	0	\
\	0	/

M( n__isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__length(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	0	0	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	0	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( length(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( U21(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

Tuple symbols:

M( LENGTH(x₁) ) =

[

]

x₁

M( U11¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

Matrix type:
We used a basic matrix type which is not further parametrizeable.

As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:

isNatIList(X) → n__isNatIList(X)
cons(X1, X2) → n__cons(X1, X2)
isNatList(X) → n__isNatList(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
take(0, IL) → U21(isNatIList(IL))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
0 → n__0
take(X1, X2) → n__take(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
activate(X) → X
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatIList(V) → isNatList(activate(V))
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
isNat(n__s(V1)) → isNat(activate(V1))
and(tt, X) → activate(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
activate(n__isNatIList(X)) → isNatIList(X)
isNat(n__length(V1)) → isNatList(activate(V1))
activate(n__isNatList(X)) → isNatList(X)
activate(n__nil) → nil
isNat(n__0) → tt
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt) → nil
U11(tt, L) → s(length(activate(L)))
zeros → cons(0, n__zeros)
length(nil) → 0
isNatList(n__nil) → tt
isNatIList(n__zeros) → tt

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U11¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U11¹(tt, x0) → LENGTH(x0)
U11¹(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U11¹(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U11¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))
U11¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U11¹(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U11¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

U11¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))

The remaining pairs can at least be oriented weakly.

U11¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U11¹(tt, x0) → LENGTH(x0)
U11¹(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U11¹(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U11¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U11¹(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U11¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( U31(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( n__s(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( n__nil ) =

/	0	\
\	0	/

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( nil ) =

/	0	\
\	0	/

M( n__isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__length(x₁) ) =

/	1	\
\	1	/

/	0	1	\
\	0	0	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

M( U11(x₁, x₂) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( length(x₁) ) =

/	1	\
\	1	/

/	0	1	\
\	0	0	/

x₁

M( U21(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

Tuple symbols:

M( LENGTH(x₁) ) =

[

]

x₁

M( U11¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

isNatIList(X) → n__isNatIList(X)
cons(X1, X2) → n__cons(X1, X2)
isNatList(X) → n__isNatList(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
take(0, IL) → U21(isNatIList(IL))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
0 → n__0
take(X1, X2) → n__take(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
activate(X) → X
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatIList(V) → isNatList(activate(V))
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
isNat(n__s(V1)) → isNat(activate(V1))
and(tt, X) → activate(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
activate(n__isNatIList(X)) → isNatIList(X)
isNat(n__length(V1)) → isNatList(activate(V1))
activate(n__isNatList(X)) → isNatList(X)
activate(n__nil) → nil
isNat(n__0) → tt
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt) → nil
U11(tt, L) → s(length(activate(L)))
zeros → cons(0, n__zeros)
length(nil) → 0
isNatList(n__nil) → tt
isNatIList(n__zeros) → tt

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U11¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U11¹(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U11¹(tt, x0) → LENGTH(x0)
U11¹(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U11¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11¹(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U11¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U11¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

U11¹(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U11¹(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))

The remaining pairs can at least be oriented weakly.

U11¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U11¹(tt, x0) → LENGTH(x0)
U11¹(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U11¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U11¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( U31(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( n__s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( n__nil ) =

/	0	\
\	0	/

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( nil ) =

/	0	\
\	0	/

M( n__isNatIList(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( n__length(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__isNatList(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	0	0	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	0	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( length(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( U21(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

Tuple symbols:

M( LENGTH(x₁) ) =

[

]

x₁

M( U11¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

isNatIList(X) → n__isNatIList(X)
cons(X1, X2) → n__cons(X1, X2)
isNatList(X) → n__isNatList(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
take(0, IL) → U21(isNatIList(IL))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
0 → n__0
take(X1, X2) → n__take(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
activate(X) → X
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatIList(V) → isNatList(activate(V))
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
isNat(n__s(V1)) → isNat(activate(V1))
and(tt, X) → activate(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
activate(n__isNatIList(X)) → isNatIList(X)
isNat(n__length(V1)) → isNatList(activate(V1))
activate(n__isNatList(X)) → isNatList(X)
activate(n__nil) → nil
isNat(n__0) → tt
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt) → nil
U11(tt, L) → s(length(activate(L)))
zeros → cons(0, n__zeros)
length(nil) → 0
isNatList(n__nil) → tt
isNatIList(n__zeros) → tt

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U11¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U11¹(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U11¹(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U11¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U11¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

U11¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))

The remaining pairs can at least be oriented weakly.

U11¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U11¹(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U11¹(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U11¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( U31(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( n__isNat(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( n__s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__nil ) =

/	0	\
\	0	/

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( nil ) =

/	0	\
\	0	/

M( n__isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U21(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

Tuple symbols:

M( LENGTH(x₁) ) =

[

]

x₁

M( U11¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

isNatIList(X) → n__isNatIList(X)
cons(X1, X2) → n__cons(X1, X2)
isNatList(X) → n__isNatList(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
take(0, IL) → U21(isNatIList(IL))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
0 → n__0
take(X1, X2) → n__take(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
activate(X) → X
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatIList(V) → isNatList(activate(V))
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
isNat(n__s(V1)) → isNat(activate(V1))
and(tt, X) → activate(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
activate(n__isNatIList(X)) → isNatIList(X)
isNat(n__length(V1)) → isNatList(activate(V1))
activate(n__isNatList(X)) → isNatList(X)
activate(n__nil) → nil
isNat(n__0) → tt
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt) → nil
U11(tt, L) → s(length(activate(L)))
zeros → cons(0, n__zeros)
length(nil) → 0
isNatList(n__nil) → tt
isNatIList(n__zeros) → tt

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U11¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U11¹(tt, x0) → LENGTH(x0)
U11¹(tt, n__take(x0, x1)) → LENGTH(take(activate(x0), activate(x1)))
U11¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U31¹(tt, IL, M, N) → ACTIVATE(IL)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
TAKE(s(M), cons(N, IL)) → U31¹(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
U31¹(tt, IL, M, N) → ACTIVATE(M)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ISNATLIST(n__take(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U31¹(tt, IL, M, N) → ACTIVATE(N)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.